Question: The first four terms of a geometric sequence are given: $-\dfrac{32}{27},\dfrac{16}{9},-\dfrac{8}{3},4, \ldots$ What is the fifth term in the sequence?
Solution: In any geometric sequence, each term is equal to the previous term times the common ratio. Thus, the second term is equal to the first term times the common ratio. In this sequence, the second term, $\dfrac{16}{9}$ , is $-\dfrac{3}{2}$ times the first term, $-\dfrac{32}{27}$ Therefore, the common ratio is $-\dfrac{3}{2}$ The fifth term in the sequence is equal to the fourth term times the common ratio, or $4 \cdot -\dfrac{3}{2} = -6$.